The isoperimetric problem

Resonance ◽  
1997 ◽  
Vol 2 (9) ◽  
pp. 65-68 ◽  
Author(s):  
Alladi Sitaram
Keyword(s):  
Isoperimetric Problem

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

AN ISOPERIMETRIC RESULT FOR THE GAUSSIAN MEASURE AND UNCONDITIONAL SETS

2001 ◽  
Vol 33 (4) ◽  
pp. 408-416 ◽  
Author(s):  
F. BARTHE
Keyword(s):  
Gaussian Measure ◽  
Isoperimetric Problem ◽  
Boundary Measure

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.

Isoperimetric Problem and Meta-fibonacci Sequences

2008 ◽  
pp. 22-30
Author(s):  
B. V. S. Bharadwaj ◽  
L. S. Chandran ◽  
Anita Das
Keyword(s):  
Isoperimetric Problem ◽  
Fibonacci Sequences

scholarly journals Exact bounds for tail probabilities of martingales with bounded differences

2009 ◽  
Vol 50 ◽  
Author(s):  
Dainius Dzindzalieta
Keyword(s):  
Random Walks ◽  
General Principle ◽  
Isoperimetric Problem ◽  
Maximal Inequality ◽  
Tail Probabilities ◽  
Bounds For Tail Probabilities ◽  
Maximal Inequalities ◽  
Exact Bounds ◽  
Natural Classes

We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\},  (1) where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximalinequalitiesP\{max 1\leq k\leq n Mk   \geq x\} \leq Bn(x).The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our caseBn(x) = supP{Mn  \geq  x}.Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R.

scholarly journals Minimality of polytopes in a nonlocal anisotropic isoperimetric problem

2021 ◽  
Vol 205 ◽  
pp. 112223
Author(s):  
Marco Bonacini ◽  
Riccardo Cristoferi ◽  
Ihsan Topaloglu
Keyword(s):  
Isoperimetric Problem
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